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Saturday, June 13, 2009

SoCG 2009: Epsilon-nets and covering/hitting problems.

The 30,000 horsepower engine of approximation algorithms is linear programming. The fact that an LP gives you a bound on the cost of an NP-hard problem has been the major force behind ever more sophisticated methods for proving approximation guarantees.

Ironically enough, given the geometric nature of an LP, this engine has not been particularly useful for geometric approximation problems. The high level reason (somewhat over-simplified) is that it is generally hard to encode specifics about geometry in a linear program. What this means is that (for example) if you want to solve a geometric covering problem and can't encode the geometry in your LP, you're stuck with the log n approximation that combinatorial set cover gives you, inspite of the fact that there are numerous examples of problems where throwing geometry in actually improves the approximation bound considerably.

To put it mildly, this is a pain in the neck. It basically means that you have to design methods that work for a specific problem, and there are only a few general purpose techniques available, many of them outlined in this survey.

But if your problem can be framed as a covering/hitting problem, then there's a general body of work that has provided better and better results, (and for an added bonus, actually has an LP-formulation). The story dates back to a paper on iterative reweighting by Clarkson, and actually goes back even earlier if you jump to the ML literature: Arora, Hazan and Kale have a nice survey on this general approach.

But I digress: the paper by Bronnimann and Goodrich put the technique in the language we're familiar with today: in short,

Given a range space that admits epsilon-nets of size (1/epsilon)f(1/epsilon), you can solve the hitting set problem to an approximation of f(OPT).

In particular, range spaces that admit epsilon-nets of size 1/epsilon admit constant-factor approximations for the hitting set problem. This led to a "formula" for approximating these problems: find an epsilon-net of an appropriate size, and declare victory. There's been much work on constructing epsilon-nets since then, partly motivated by this.

Another angle to this story was put forward by Clarkson and Varadarajan in their 2005 SoCG paper: the brief summary from that paper was:

Given a range space in which you can bound the complexity of the union of a random subcollection of size r by rf(r), you can find an epsilon net of size (1/epsilon)f(1/epsilon).

This approach gave another tool for finding good approximations for covering problems (and in particular gave a nice constant factor approximation for a thorny guarding problem).

Given a range space in which you can bound the complexity of the union of a random subcollection of size r by rf(r), you can find an epsilon net of size (1/epsilon)log f(1/epsilon).

Although this doesn't improve results in the constant factor regime, where f() is a constant, it makes a significant difference when f() is non-constant.

The interesting story here is that Kasturi's result was simultaneous with another paper by Aronov, Ezra, and Sharir that just appeared in STOC 2009. They get the same bound without the technical details that limit his result, however their methods are quite different.

Returning to LPs, what I failed to mention is that the epsilon-net construction technique used by Bronnimann and Goodrich can be seen as a byproduct of an LP-algorithm, as was shown by Even, Rawitz and Shahar, and Philip Long before that (thanks, Sariel)